Optimal. Leaf size=51 \[ -\frac{\left (a+b x^n\right )^{5/2} \, _2F_1\left (1,\frac{5}{2}-\frac{2}{n};-\frac{2-n}{n};-\frac{b x^n}{a}\right )}{2 a x^2} \]
[Out]
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Rubi [A] time = 0.074685, antiderivative size = 61, normalized size of antiderivative = 1.2, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{a \sqrt{a+b x^n} \, _2F_1\left (-\frac{3}{2},-\frac{2}{n};-\frac{2-n}{n};-\frac{b x^n}{a}\right )}{2 x^2 \sqrt{\frac{b x^n}{a}+1}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^n)^(3/2)/x^3,x]
[Out]
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Rubi in Sympy [A] time = 6.94971, size = 49, normalized size = 0.96 \[ - \frac{a \sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, - \frac{2}{n} \\ \frac{n - 2}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{2 x^{2} \sqrt{1 + \frac{b x^{n}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b*x**n)**(3/2)/x**3,x)
[Out]
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Mathematica [A] time = 0.139143, size = 102, normalized size = 2. \[ \frac{4 \left (a+b x^n\right ) \left (4 a (n-1)+b (n-4) x^n\right )-3 a^2 n^2 \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},-\frac{2}{n};\frac{n-2}{n};-\frac{b x^n}{a}\right )}{2 (n-4) (3 n-4) x^2 \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^n)^(3/2)/x^3,x]
[Out]
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Maple [F] time = 0.062, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3}} \left ( a+b{x}^{n} \right ) ^{{\frac{3}{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b*x^n)^(3/2)/x^3,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{n} + a\right )}^{\frac{3}{2}}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^(3/2)/x^3,x, algorithm="maxima")
[Out]
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^(3/2)/x^3,x, algorithm="fricas")
[Out]
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b*x**n)**(3/2)/x**3,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{n} + a\right )}^{\frac{3}{2}}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^(3/2)/x^3,x, algorithm="giac")
[Out]